When you hard-cut in the VSE, does the current frame end up on the left or right side of the cut?
I have here 15 frames, each one with a number on it.

When I followed what you did, frame #10 got cut into it's own strip, which can be deleted:



This are the surrounding frames:





I don't know what you are doing wrong, because it works for me.

What this means is that the frame moves to the right of the cut.

Based on this, what you should do is:


Move to the frame you want to cut.
Enter Shift + K
Move 1 frame to the right
Enter Shift + K again
Select the middle frame and delete it.
Select the next strip and move it 1 frame to the left.